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The Connecticut League of Conservation Voters is a non-profit organization working to educate voters and elected leaders on state environmental issues. CTLCV (and CTLCV Education Fund) works with dozens of groups to develop and support good environmental policies at the state legislature. We publish an annual environmental scorecard grading legislators on their environmental voting records, endorse candidates for state office who will make the environment a top priority, and help to get conservation-minded people informed and involved. For additional information, please visit both web sites at www.conservationeducation.org and www.ctlcv.org.
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About CTLCV
The Connecticut League of Conservation Voters (CTLCV) is a bipartisan, statewide nonprofit organization dedicated to protecting our environment by making it a priority for our elected leaders. As a legislative watchdog we work in concert with environmental advocacy groups across CT to identify and highlight important bills impacting our air, water, wildlife, open space and health, and turn those priorities into legislative success at the state and local levels. CTLCV is committed to ensuring that disadvantaged communities benefit from equitable, targeted, and community-driven investments in the transition to a clean energy economy. Through advocacy and education, we keep lawmakers informed of the latest issues, track key legislation, inform elected officials of the pro-environment position, hold lawmakers accountable in our annual Scorecard, and elect candidates willing to fight for clean air, clean water, and climate action.
CTLCV is an equal opportunity employer committed to working toward a just, equitable, diverse, and inclusive environmental movement, and seeks to hire staff and contractors that reflect the diverse communities and perspectives that make up Connecticut. We are a growing organization that offers a vast array of experience with opportunity for advancement and encourage people of all diverse backgrounds to apply.


We are less than 30 minutes drive from the airport, which is served by RYAN AIR and FLYBE. Flights are available from many British airports.
From Paris to Limoges railway station takes about 3 hours.
We are about 440 miles from CALAIS on good roads. From LE HAVRE, CAEN, ST. MALO, DIEPPE etc. the journey is 300 to 400 miles. We are situated via the A20 motorway exit 28. Take N21 towards PERIGUEUX, then the D66 between CHALUS and GORRE.


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The arts and crafts of a truly stylish break from Devon to Scotland
Millennials love to think they discovered everything and have even come up with a whizzy term – cottagecore – to describe an appreciation of the aesthetic beauty and romance of rural life.
Fuelled by the power of social media, the cottagecore trend is currently sweeping the United States, but this idealisation of style has been around in Britain for more than 150 years thanks to the Arts and Crafts movement. It was driven by William Morris and his followers, who set about championing the quality of design and celebrating traditional workmanship.
Hipsters before their time, every detail mattered; the Arts and Crafters wanted to design everything from door knobs to drawers, often using flowers and plants as inspiration. Railing against industrialisation and mass-produced furniture, pioneers headed to the Cotswolds and set up guilds to protect ancient trades. Morris's Cotswolds home, the 16th Century Kelmscott Manor (sal.org.uk/kelmscott-manor), is currently closed for renovation, but there is still plenty for devotees to see when travel restrictions are lifted.
Winsford Cottage Hospital, Devon
Winsford Cottage Hospital, pictured, is located in the village of Beaworthy in Devon. It was once a local hospital treating First World War soldiers
Conservation charity the Landmark Trust has highlighted the Arts and Crafts trend with its new property in the village of Beaworthy.
Built in 1900 and designed by the eminent Arts and Crafts architect Charles Voysey, this was once a local hospital treating First World War soldiers wounded on the front line.
Voysey created everything from the motifs of hearts and trees to the windows and the doors. He also designed the wards so that patients could look out on to the sunny, south-facing garden.
Everything has now been serenely updated, with proper kitchens and tasteful bathrooms.
The property sleeps up to six people, and four-night stays cost from £503 (landmarktrust.org.uk).
Gravetye Manor, Sussex
One of the bedrooms at Gravetye Manor in East Grinstead, which dates back to the 16th Century. It was transformed into a hotel in 1958
Gravetye Manor is ancient – it dates back to the 16th Century – and in 1885 it was bought by William Robinson. Originally a gardener's boy, Robinson became one of the most influential horticulturalists of his era and Gravetye was his home until he died in 1935.
The manor, near East Grinstead, was transformed into a hotel in 1958, and today the rooms still adhere to Arts and Crafts principles: they are grand yet serene.
Guests can also sign up for daily tours of the meadows, kitchen and flower gardens.
B&B doubles cost from £295 a night (gravetyemanor.co.uk).
Perrycroft, Herefordshire
Perrycroft in Malvern, which was built by Charles Voysey in 1893. It has been refurbished to contain plenty of modern conveniences
On a smaller scale, Perrycroft, near Malvern, was built by Charles Voysey in 1893 for John Wilson, an MP and industrialist. It's a private home but it is open to the public between May and September. There are three holiday cottages on the estate too, including The Lodge, which was also built by Voysey.
Although The Lodge has been refurbished to contain plenty of modern conveniences, much care has been taken to maintain Voysey's vision, thanks to whitewashed walls, green painted woodwork, red curtains, and antique furniture and fittings.
The property has three bedrooms and sleeps up to six people. A four-night stay costs from £585 (perrycroftholidaycottages.co.uk/the-lodge).
Llangoed Hall, Wales
Llangoed Hall was originally an ancient manor house near Hay-on-Wye. It has an art collection that includes works by Augustus John and James McNeill Whistler
Originally an ancient manor house near Hay-on-Wye, Llangoed Hall was rebuilt as a mansion in 1912 by Clough Williams-Ellis. He was fascinated by the notion of village life and would later go to build the Italianate village of Portmeirion in North Wales.
Keeping the Arts and Crafts flame alive, in the 1980s Bernard Ashley, husband of designer Laura Ashley, turned Llangoed Hall into a country house hotel, with kitchen gardens and an art collection that includes works by Augustus John and James McNeill Whistler.
The property has since changed ownership but the ethos remains the same. B&B doubles cost from £160 a night (llangoedhall.co.uk).
Standen House, West Sussex
Inside Standen House, which was built in 1891 by Philip Webb, an influential Arts and Crafts designer
Philip Webb was another influential Arts and Crafts designer and in 1891 he built Standen House, near East Grinstead, for a wealthy family.
Now owned by the National Trust, the place is an Arts and Crafts time capsule, with William Morris furnishings and wallpaper, and you can still see the original electric light fittings. Guests staying in the property's Morris Apartment can explore the gardens after visitors have left for the day.
The one-bedroom apartment sleeps up to four people and features an elegant sitting room and a light and airy kitchen. A two-night stay costs from £494 (nationaltrust.org.uk/holidays/the-morris-apartment-sussex).
Rodmarton House, Gloucestershire
This house was built by Ernest Barnsley, a follower of Morris, and is still owned by the family who commissioned it in 1909.
Rodmarton is packed with early 20th Century treasures, including furniture and pottery by Alfred and Louise Powell, wall hangings by Hilda Benjamin, lead and brass designed by Norman Jewson, and ironwork by Fred and Frank Baldwin and Alfred Bucknell.
Visitors can take tours of both house and garden, and craft events take place regularly (rodmarton-manor.co.uk).
Hidcote, Gloucestershire
The beautiful gardens at Hidcote in Chipping Campden, which are among the most charming in Britain
The 17th Century property was bought by Lawrence Johnston in 1907 and he quickly set about putting into practice all that he had learned from books by Arts and Crafts devotees.
Today, the gardens at Hidcote, in Chipping Campden, are among the most charming in Britain – a series of colourful and intricately designed outdoor 'rooms'.
The site is now owned by the National Trust (nationaltrust.org.uk/hidcote) and continues to attract 175,000 visitors a year. Also in Chipping Campden is Court Barn (courtbarn.org.uk), a museum that showcases the Arts and Crafts period.
The William Morris Gallery, pictured, in Walthamstow, East London, houses the largest collection of his designs
However much the Arts and Crafters loved the countryside, they couldn't escape London entirely. The William Morris Gallery is housed in Morris's very grand childhood home in Walthamstow, East London, and has the largest collection of his designs (wmgallery.org.uk).
Following their marriage, Morris and his wife Jane went on to live at Red House in Bexleyheath, Kent.
Built in 1860 by Morris and Philip Webb, it's another jewel, with paintings and murals by Pre-Raphaelites Edward Burne-Jones and Dante Rossetti and a touch of utopianism – the servants' rooms are unusually light and airy (nationaltrust.org.uk/red-house).
Martin Randall Travel has been conducting Arts and Crafts tours of the Cotswolds since 2016, offering guests a chance to visit museums, churches and private houses.
A four-night tour in September starts at £1,890pp including all transport and accommodation, and most meals. The itinerary includes visits to a branch of the world-famous Ashmolean Museum in Broadway, which includes a collection on vernacular British decorative arts, and the Museum and Art Gallery in Cheltenham, which has a nationally important Arts and Crafts collection (martinrandall.com/arts-and-crafts-in-the-cotswolds).
Meanwhile, Historic Houses also arranges tours of privately owned Arts and Crafts houses, including Owlpen Manor in Gloucestershire (historichouses.org).
Hill House in Helensburgh, near Loch Lomond. It was built by renowned architect Charles Rennie Mackintosh and his wife Margaret
Renowned architect and designer Charles Rennie Mackintosh and his wife Margaret built Hill House in Helensburgh, near Loch Lomond, between 1902 and 1904 for publisher Walter Blackie.
The house is considered to be his domestic masterpiece – Mackintosh and his wife built the furnishings too and even gave the owners instructions on what colour flowers should be put in to vases.
However, the exterior materials used have not withstood the test of time, so the National Trust for Scotland is embarking on a ten-year restoration project. A protective steel structure has been built over the house so that work can continue away from the elements (nts.org.uk/visit/places/the-hill-house).
TREASURE TROVES OF THE LAKES
Troutbeck Church in Cumbria, which has Arts and Crafts interiors
They were built largely as holiday homes for Manchester industrialists, and nowhere in Britain has a better collection of Arts and Crafts mansions than the Lake District.
Even though many of them are still private homes, you can visit Blackwell (blackwell.org.uk). Overlooking Windermere, it was built in 1898 by Baillie Scott for a brewing magnate.
It's an Arts and Crafts treasure trove, with peacock friezes, an all-white drawing room and leaf-shaped door handles.
The garden is just as important, with a series of terraces framing the water below. Blackwell's craft connections are kept current with a shop that showcases contemporary artists, particularly ceramicists.
There are many churches in the area worth seeing too – those at Staveley and Troutbeck have Arts and Crafts interiors, including stained-glass windows by Morris and the Pre-Raphaelites.
Brantwood was the home of the art critic John Ruskin. Overlooking Coniston Water, it has eight gardens to explore plus a museum containing his art and belongings.
If you want to soak up the view a bit longer, you can even stay here. The Lodge sleeps up to nine while the Eyrie is a one-bedroom apartment overlooking the water. Two-night stays start at £247 (brantwood.org.uk).
Birkdale House in Bowness- on-Windermere shows off its Arts and Crafts roots with fireplaces and stained-glass windows, although the cellar's transformation into a cinema room may not fit in entirely with Morris's views on modernity.
Three-night breaks at the property, which sleeps ten, start at £3,675 (birkdalewindermere.co.uk).
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redpajama_set_name: RedPajamaCommonCrawl


Rachel Gordon on "Tapping into Educational Distribution Part 2"
Yesterday, Rachel Gordon shed some light on how you might make your film viable for the Educational Market. Now as much as we all hope to make a living by making films, I don't think that is why most filmmakers enter the field. And as thrilling as self expression is, I often hear filmmakers cite another reason for the creative spark: they want to facilitate change. Today, Rachel provides examples of how the process of preparing for the Educational Market can also precisely do that higher goal of moving us towards a better world.
Adding to the idea of using media in education, this post will provide a broad view of integrating media with community change, as well as concrete examples of success.
If you ever watched a film in high school or college, or went to a screening at a local community center, you've already experienced media having an impact on a non-theatrical audience. Here is the short description of how that comes about, as well as specific situations from clients I've worked with.
The Collector of Bedford Street centers on a retirement-aged Jewish man with an intellectual disability who spends his days collecting money for a variety of charities that request him to do so. It shows him being an active participant in the community, and the mutual care-giving relationship between himself and his neighborhood – he's able to continue collecting money for charities, and his well being is sustained through surrounding efforts.
Now take one agency, perhaps one of the ones involved with the care of your subject, and get their perspective on your finished product. Show them a rough cut to get them on board and get ideas about who needs to know about what you're doing. Be willing to give out preview copies to one or more of these organizations in exchange for feedback. Use their feedback in order to forward it to others who have similar interests.
How do you use the film?
What are some of the reactions you've seen?
Who do you think should see this?
To get specific, Collector has been used by Kiwanis International to help teach youth about the importance and joys of community service. Inspired by that, for the past couple of years, Roger Williams University has used Collector's story, including filmmaker Alice Elliott as a speaker, at their student orientation to help demonstrate the positive change that results from participating in community service activities.
Every time a copy of your film leaves your hands, see who it went to and note what type of organization they are coming from. Write them thanking them for support and seek their feedback to build on, and quote as well.
When your film is showing, contact those in your interest groups to notify them, assuming you can invite others to see it. Or, even if you can't, send out information about why it's being shown in that community to those same parties. This would include public libraries, colleges/universities, local non-profit advocacy groups, etc. They may not be able to attend, but the screening provides community respect and they may at least check your website in interest.
Another concrete example was an event I recently coordinated for the Massachusetts Institute of Technology. The Women and Gender Studies Division decided to work with their Student Disability Services Department to host a screening of the film Body & Soul: Diana & Kathy. They wanted the subject, Diana Braun, to speak at the event, but Diana was overseas promoting disability self-advocacy in Uzbekistan through the American Documentary Showcase. In order to make a more powerful event, I connected with the local Arc in Massachusetts – a disability advocacy group – who spoke alongside me at the event. After screening the film a lively discussion ensued about how to help ensure the independence of people with disabilities, and what any individual could do in support of disability rights.
Both of these films are under 1 hour long and both have been making significant impact in a variety of communities, and earning income in the process. If they can do it, any film can do if it you're willing to put in the time and effort.
In the over-an-hour category is In Good Conscience, about a Catholic nun non-violently fighting for gay rights, with which we've managed to create public events at universities and churches. Sister Jeannine speaks with the film, along with filmmaker Barbara Rick, and it gets used as a tool to discuss bias, bullying, LGBT inclusion, and religious integration.
Another film over an hour, which also had a theatrical run, that I'm about to attend the American Library Association in support of is The Most Dangerous Man in America: Daniel Ellsberg and the Pentagon Papers. The film is screening during the conference, and Daniel Ellsberg is also speaking there in an entirely separate program.
These engagements take some time to plan, as these institutions need to prepare budgets well in advance in order to prepare travel arrangements, technical support, etc. When an event happens at a university, chances are the institution will take an extra step of coordinating multiple activities with different departments to get their monies' worth – such as the film or journalism department, schools of Social Work, Student Activities, etc.
It's also worth noting that films can have a long lifespan in the educational environment, where acquisition and usage are based on theme and research areas. So a film isn't discarded or forgotten because it's already been out for 2 years, it can find strong validity in the classroom for over a decade. Creating and maintaining an educational and advocacy agenda can build you a worthwhile audience.
Rachel Gordon on "Tapping into Educational Distribution"
For Indie Film to thrive, a producer must consider all revenue streams from the beginning. Of course not every project, is applicable to every opportunity, but nonetheless, you want a box to check for each. The Educational Market is one platform that goes unchecked for many filmmakers. Let's change that!
Filmmaker, and expert on this field, Rachel Gordon graciously offered to share what she knows of this field with you.
I began working with non-theatrical distribution at the National Film Board of Canada over 10 years ago. As a filmmaker, the experience of finding new markets for films that were under an hour, and even animated, was exhilarating as I'd worked on so many films that would never see the inside of a theater.
Educational distribution is broadly defined as any usage of media that is not consumed in a traditional theater, or home video setting. Examples include, but are not limited to: classrooms – both K-12 and colleges, museums, non-profit/advocacy groups, etc. It is not an exact science, and it often takes longer to start seeing returns than people have patience for. If you stick with it long enough, though, you'll connect directly with an invested audience that will keep up with your projects.
So discussed here is how to make your film an educational tool, no matter what its length or genre, and hopefully pay a few bills in the process. This should not be relied on as your only form of income. Academic environments take longer because people plan courses ahead of time so expect this outreach process to take a minimum of 6 months, up to a year, to hit solidly.
1. During DVD creation, provide chapters of your film that are 5-7 minutes long. Don't randomly pick the timing, use whole thought segments.
2. Disability accessibility features such as closed (or open) captioning and audio description may seem like "extras" but are becoming more necessary as state agencies and educational facilities adopt ADA (American with Disabilities Act) specifications into law.
3. Study guides are highly useful, as your way of helping any audience understand what they are supposed to get out of watching your film. Educators appreciate these as they often lack preparation time and energy.
4. Be prepared to process orders in any way that is convenient for your customers – including check, purchase order, credit card. If you make it hard to buy or use your film, people will lose patience and not ask for help.
5. Be flexible about how the content gets delivered. Current options include DVD, streaming, digital download, closed-circuit/institutional television, and the right for media to be accessed from a central server.
6. Be creative with photos and artwork. DVD covers should include pictures, synopsis, and quotes. If you hand someone a clear case, it's going to look bland, uninteresting, and less reliable as an educational source.
How would you use the film? Who do you think should see this? What social action goals does this film serve?
Every time a copy of your film leaves your hands, see who it went to and note what type of organization they are coming from. Write them thanking them for support and seek their feedback to build on.
It's also important to send review copies to the top educational publications – almost every media librarian in the country subscribes to them – university and public libraries alike – and they also peer-review the materials. Give about 3-6 months time for them to get to it. These publications include (but again, aren't limited to): Video Librarian, Educational Media Reviews Online, Library Journal.
There are academic studies dedicated to any subject you can think of, and they all use media as a tool with which to engage their students because they realize that students are consuming online video content. The academic world extends beyond the classroom to include conferences, publications, and professional development.
It is, and seems like, a long process, but what you get from all of this work are people who do come back and want to continue to use your media to help with their programming. What is also amazing is how positive and supportive educational environments are. Librarians who use and promote your work respect copyright issues, talk on a variety of listservs about content that they find helpful, and are generally quick to respond and answer questions about what is going on in their worlds.


Pre-sales/technical solutions consultant with over 20 years of industry experience leading teams to define complex software solutions and benefits that match client's business needs. Harold is a European subject matter expert in operational prescriptive and predictive Decision Management based on Corticon Business Rules, the DataRPM Cognitive Anomaly Detection & Prediction solution and Sitefinity's Digital Experience Cloud (DEC), but also has a thorough understanding of the other solutions in the Progress software portfolio.
He is responsible for liaising with new, existing customers, distributors, ISV & implementation partners and (Cloud) service providers throughout EMEA to understand business needs, help define and present solution offerings and drive revenue opportunities. Harold has been with Corticon for more than 6 years before joining Progress in 2012 with prior roles in IT Development, Partner Support, Marketing, Business Development and Presales Consulting for various international IT software vendors. He lives in the Netherlands, is married with 2 children and speaks 4 languages.
Progress Corticon is a powerful, easy to use business rules management system (BRMS) that tames the complex policies and regulatory rules enterprises face without user, performance, platform or language limitations. Automating decision management with Corticon BRMS enables businesses to strengthen their applications and keep pace with change. In our upcoming release, we up the ante even further by supporting REST data connectivity via the Progress DataDirect Autonomous REST Connector - an intelligent, customizable data connectivity solution enabling businesses to build and deploy enterprise-grade data connectors in minutes without coding. In this session, we'll walk through examples of how this combined functionality empowers customers with seamless access to the expanding sources of data that drive their business.


I have been studying Tasawwuf for about seven years and I have found it to be an incredibly fulfilling and enriching spiritual journey. During this time I have deeply pondered and explored the relationship between Tasawwuf and Islam. Academically one could classify various groups that refer to themselves as "Mutassawufen" ranging from Islamic Spirituality to non-Islamic Spirituality or in other words groups that have little to no affiliation with Islamic tenants and values. In the earliest part of my journey I would even say that I was dubious towards traditional Islam, given the fact that most my initial impression of Islam came from the western media which tended to portray it as a violent and oppressive religion. In my studies however, I have come to believe that Islam is actually a peaceful religion when practiced and understood properly. Also I have come to learn that Tassawuf is in fact the inner essence of Islam and in fact the two are inseparable. My Mutassawufen teacher has said that Islam is like the bark of the tree, and Tassawuf is like the sap. The bark "outer teachings" protects the sap or the "inner essence" of the faith.
I believe that Tassawuf is intrinsically a part of what makes Islam a healthy and dynamic faith and I would personally say that the horrors that we see committed in the world today in the name of Islam are largely an effect of the Islamic faith having forgotten the inner principles of Islamic Spirituality. In fact many of the Wahhabi movements have declared Tassawuf to be a heretical innovation of the faith. Not surprisingly, it's the radical fringes of these same movements that end up turning into ISIS or other Islamic terrorist groups. This is why I feel it is of the highest of importance to revive the integral Tassawuf teachings to bring Islam back into balance.
So what then is Islam?
To start I would like to explain the etymology of the word Islam. For example you have probably heard the Muslim greeting "Asalaam Alaikum" which means, "Peace be upon you." So the word Salaam means "peace" in Arabic. "As-Salaam ", "The Peaceful" is also one of the 99 Names of God in Islam. So now when we look at the word "Is-lam" we can see it is partly derived from the word for peace "salaam." Traditionally Islam has been translated as "Surrender". However when you extract the deeper meaning from the etymology you can see that Islam really means, "The Peace that Comes through Surrendering To God." So from its essence you can see the word Islam is actually about peace. The idea here is that when one lives a life that is in harmony with divine will, the natural state of being that results is peace. Hamza Yusuf, an American Islamic scholar pointed out that ironically, all of these people advocating for forming an "Islamic State", are not themselves in a state of" Islam" or "inner peace."
So then what then is Tassawuf?
Within Islamic Theology there are various sciences, such as the science of "fiqh" or Islamic Jurisprudence. There is also the science of aqeeda or belief which uses rational proofs and argumentation to derive understanding as to the nature of God. Tassawuf then can be seen as one of the sciences within Islam, and those who understand what it is would agree it is the most important science. Tassawuf is the science of spiritual excellence or in other words it is a spiritual discipline for removing from the blameworthy qualities of the ego, such as arrogance, envy, anger, greed, stinginess etc from one's character. It is a method of purifying one's inner most intentions to seek to only please Allah or God. Unlike the other disciplines within Islam, Tassawuf is a path of tasting and therefore it cannot be learnt in a book but rather it must be implemented and experienced within one's life.
The goal of Tassawuf is to reach a spiritual state of Union with God known as "Fana" or "Annihilation". In this sense annihilation means the annihilation of one's egoic consciousness into the supreme divine consciousness . This is achieved through the purification of one's inner most heart of the aforementioned blameworthy egoic qualities. It is said within the Islamic tradition that the heart of the believer is the "Throne of Allah". So Allah can only really be found through journeying through ones innermost heart where the aspiring seeker can experience a state of union with God. It is also said there are 70,000 veils between the human being and God and it is through the purification of these veils that one begins to experience the divine. During these spiritual tastings, one can travel from intellectual understandings of God as being Merciful, Peaceful, Loving, etc, to actually experiencing these qualities of God within the depths of one's soul. It is said that there is a huge difference between knowing about God and Knowing God. It is the latter that the Sufi is interested in.
Although the topic of understanding the tenants of Islam is far to vast to be covered in this essay, what I can say is that it absolutely clear that Islam forbids killing civilians, and it sets very strict rules for engaging in warfare. According to BBC website on Islam there are only 4 valid reasons for war in Islam: 1. Islam allows war in self-defense (Qur'an 22:39), 2. to defend Islam (rather than to spread it), 3. to protect those who have been removed from their homes by force because they are Muslims (Qur'an 22:40), and 4. to protect the innocent who are being oppressed (Qur'an 4:75).
So you can see that the Islamic Terrorist groups do not have any legitimacy in their use of violence especially against civilians. Also, since suicide is prohibited in Islam, that also rules out suicide bombing. The truth is that human beings have been using religion or any other doctrine such as communism or fascism to justify horrific crimes against humanity. Groups ISIS in this sense are no different. If you want a more in depth understanding of where ISIS came from I highly recommend the Huffington Post article in footnote 1 called "You Can't Understand ISIS If You Don't Know the History of Wahhabism in Saudi Arabia" by Alistair Crooke.
The second quote can be found almost verbatim in the Jewish text the Talmud.
"Whoever destroys a soul, it is considered as if he destroyed an entire world. And whoever saves a life, it is considered as if he saved an entire world." — Mishnah Sanhedrin4:9; Yerushalmi Talmud, Tractate Sanhedrin 37a.
In conclusion I feel a deep calling to help heal the Islamic faith by helping to return the understanding of Tassawuf both to Muslims and non-Muslims. Initially embarking on the Tassawuf path, I resisted referring to myself as being a Muslim for fear of being mis-understood and labeled as something which represents the opposite of what I believe in. I however feel that as part of my responsibility as one who practices this faith is to refer to myself as Muslim and to do the best to represent what I believe that to mean: one who has peacefully surrendered to God.
A great article closely related.


Have not really heard very many good comments about this vise. Some of the early run were rough and had problems, so they might be re-engineering them a bit before going to a wider release (just a guess). Personally, in that price range, I can think of several other vises I'd rather buy.


Believe it or not but, in the 1940s, the Leicester area boasted no less than 25 cinemas.
This is according to a copy of a cinema listings page carried in the Leicester Mercury, which was sent in by reader Ray Newcombe, who lives in Aylestone.
The listings included the name of the cinema, which films were being shown and their start times and certificate.
These were the days when films were classified into three categories: U standing for Universal and suitable viewing for children; A standing for Adult, with some councils ruling that children must be accompanied by an adult; and H for Horror.
Some councils ruled that only those aged 16 or over could be admitted to these movies.
The H certificate was added in 1932 and remained in force until 1950, when it was replaced by the X certificate, which was enforced by all councils and ruled that only those aged 16 and older could be admitted.
Of course, these were the days before multi-screen cinemas when many just had one screen, exceptions to this including the Savoy on Belgrave Gate and the Trocadero on Uppingham Road which each had two.
Some of the films of the day make interesting reading.
The Lawn, at Birstall, was showing Cover Girl, starring Rita Hayworth, while the Sovereign, on Woodgate, and Shaftesbury on Overton Road were advertising Magnificent Doll starring Ginger Rogers and David Niven.


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Rolf Kleinert (* 24. November 1911 in Dresden; † 20. Januar 1975 in Berlin) war ein deutscher Dirigent.

Leben und Werk
Kleinert, Sohn eines Porzellanmachers, studierte von 1931 bis 1933 an der Orchesterschule der Sächsischen Staatskapelle Dresden Violine, Klavier und bei Fritz Busch Dirigieren. Weitere Lehrer waren Kurt Striegler, Hermann Kutzschbach und Johannes Schneider-Marfels. Er spielte auch Oboe und Trompete. Er arbeitete zunächst als Kapellmeister am Theater Freiberg und als musikalischer Oberleiter und Dirigent der Sinfoniekonzerte am Stadttheater Brandenburg/H. 1941 musste Kleinert seine Laufbahn unterbrechen, da er zur Wehrmacht eingezogen wurde. Bei Kriegsende geriet er in französische Gefangenschaft. 1947–1949 leitete er das Mitteldeutsche Rundfunkorchester im Sender Leipzig. 1949–1952 übernahm er die Aufgabe des Musikdirektors am Theater Görlitz. Die DDR-Erstaufführung der polnischen Nationaloper "Halka" wurde von ihm initiiert und aufgeführt.

1952 begann eine langjährige Zusammenarbeit mit dem Rundfunk-Sinfonieorchester Berlin. Erst als 1. Dirigent neben dem Chefdirigenten H. Abendroth. Nach dessen Tod 1956 leitete Kleinert das Orchester. 1959 wurde Kleinert in der DDR zum Generalmusikdirektor ernannt und übernahm als Chefdirigent das Rundfunk-Sinfonieorchester Berlin. 1960 erhielt er den Professorentitel. Durch den Bau der Berliner Mauer 1961 verlor das Orchester ein Drittel der Musiker. Das Orchester stand kurz vor der Auflösung. Kleinert und Hanns Eisler kämpften vehement für das Bestehen des Orchesters. Durch sein intensives Engagement bei der Suche nach geeigneten Musikern, gelang es, das Orchester wieder spielfähig zu machen und seinen spezifischen Klang zu erhalten. Konzertreisen mit dem Orchester führten Kleinert nach Italien, England, Westdeutschland, Sowjetunion, Polen, Bulgarien. Er wurde immer wieder vom National Orchester Chile, den Philharmonien Belgrad, Leningrad (Petersburg), Moskau, Tschechien, Riga, Budapest, Bratislava, Kairo und dem RSO Helsinki zu Gastdirigaten eingeladen.

Kleinert erhielt den Vaterländischen Verdienstorden und den Nationalpreis der DDR.

Im Jahr 1972 erkrankte Kleinert und musste das Dirigieren aufgeben. Er verstarb 1975 und wurde auf dem Striesener Friedhof in Dresden beigesetzt.

Im Deutschen Rundfunkarchiv liegen ca. 800 von ihm dirigierte Aufnahmen.

Bedeutung
"Kleinert war ein Dirigent der 'werktreuen' Schule, aus der Tradition Leibowitz, Toscanini und Scherchen kommend. Tempo, Akkuratesse, Klarheit und tänzerische Eleganz wusste er jeder Partitur zu entlocken." Dr. M. Meyer, in Rundfunk Sinfonieorchester 1923–1998. Eine vorzügliche Schlagtechnik, großer Klangsinn und ein absolutes Gehör waren Kleinerts wesentliche Eigenschaften.

Diskographie (Auswahl)
 Beethoven: Konzert für Klavier und Orchester Nr. 2 B-Dur, op.19, Sol. D.Zechlin/RSOB 1965
 Beethoven: Sinfonie Nr. 7 A-Dur, op.92/RSOB 1965
 Brahms: Konzert für Violine, Violoncello und Orchester, a-Moll, op.102 Sol.Garay, Aldulescu/RSOB 1966
 Dessau: Sinfonie Nr. 2 (1943)/RSOB 1962
 Dessau: Bach-Variation/RSOB 1970
 Elgar: Enigma-Variationen op.36/RSOB 1971
 Fritz Geißler: Kammersinfonie "45"/RSOB 1968
 Gerster: Konzert für Klavier und Orchester, Sol.S, Stöckigt/RSOB 1959
 Paul Hindemith: Philharmonisches Konzert (Variationen für Orchester 1932)/RSOB 1968
 Rolf Liebermann: Concerto für Jazzband und Sinfonieorchester/RSOB u. Tanzorchester des Berliner Rundfunk 1966
 Lutoslawski: Konzert für Orchester/RSOB 1967
 Méhul: Sinfonie Nr. 1/ RSOB 1954
 Dmitri Schostakowitsch: Konzert für Klavier, Trompete und Streichorch.op.35/S.Kootz,Krug/RSOB
 Schumann: Sinfonie Nr. 3 (Rheinische) RSOB 1971
 Spies: Violinenkonzert (1953) Sol.E.Morbitzer/RSOB 1955

Literatur
 Rundfunk-Orchester und -Chöre gGmbH Berlin (Hrsg.): Rundfunk Sinfonieorchester 1923–1998. Berlin 1998
 Meyers Neues Lexikon, Bd. 7. VEB Bibliographisches Institut Leipzig, 1973
 Landesverwaltungsamt Berlin

Weblink

Einzelnachweise

Dirigent
Klassische Musik (DDR)
Träger des Vaterländischen Verdienstordens in Silber
Träger des Nationalpreises der DDR III. Klasse für Kunst und Literatur
Deutscher
Geboren 1911
Gestorben 1975
Mann

redpajama_set_name: RedPajamaWikipedia


I love the quality and craftsmanship of my Memorial Bracelet. I have always wanted to wear a Memorial Bracelet for a long time now. I am a combat veteran myself so knowing the sacrifice it takes to serve is close to my heart. I always looked for a bracelet (in army/navy stores) but never found one that stood out. A few months ago I learned that the owner of a local business had a father that was still MIA from the Vietnam war. His mother even started a "Where is LT Joe Dunn?" campaign and recently published a book about the experience. The owner and his wife are such good genuine people I felt it appropriate to make a bracelet to honor his father's sacrifice and remind me that he will come home.


text:
Ethel E. Mackiernan was the first woman appointed to a presiding judgeship in the Commonwealth of Massachusetts.

Mackiernan attended Portia Law School.

Mackiernan began her legal career in Boston in 1918 and became attorney for the Nantucket Institution for Savings. She was the first woman to be appointed to a full-time judgeship. In 1935, she was appointed by Governor Joseph B. Ely as a judge in the Nantucket District Court. In this position she became the first presiding judge of any court in Massachusetts.

Mackiernan was an incorporator of the Norfolk Varnish Co. and assistant treasurer of the Jamaica Plain Cooperative Bank. She lived at 240 Babcock Street in Brookline, Massachusetts. She was a member of the Appalachian Mountain Club.

See also
List of first women lawyers and judges in Massachusetts

References

American women lawyers
People from Brookline, Massachusetts
Massachusetts state court judges
American women judges


text:
Why Clinton Supporters Wear White
Story from Celebrity Style
The Inspiring Reason Hillary Clinton Supporters Are Wearing White To The Polls
Elizabeth Yuko
Photo: Alex Menendez/REX/Shutterstock.
Despite being more than two months after Labor Day, expect to see a lot of white at the polls on Election Day. According to the Boston Globe, some supporters of Hillary Clinton are planning to wear white on Tuesday in homage to the suffragists of the early 1900s who fought for women's right to vote in America. After Clinton wore an all-white pantsuit to accept the Democratic party's nomination in July, the New York Times indicated that she likely wore the color to pay tribute to the suffrage movement and the National Women's Party, whose colors were white, purple, and gold. She also wore white for the last debate with Republican nominee Donald Trump. This tradition may have started when Geraldine Ferraro wore white when she accepted her nomination as the first female vice presidential candidate at the 1984 Democratic National Convention. For many who back Clinton, wearing white will celebrate the historic nature of her campaign — one that has, at times, been misogynistic in tone. The trend appears to be spreading organically, without official coordination from the Clinton campaign. Some are taking to social media, posting photos of women clad in white headed to early voting with the hashtag #WearWhiteToVote to Facebook and Twitter.
"It's a wonderful tribute to a historic campaign and what will hopefully be a wonderful presidency," Michelle Gajda, a college instructor in Tampa, FL, told the Boston Globe.
Clinton Wear White To Vote Hashtag Election Day Polls
Celebrity Style • Fashion • Hillary Clinton • The Latest
written by Elizabeth Yuko
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text:
GCGI Journal
Globalisation for the Common Good Initiative Journal
About GCGI
About GCGI Journal
GCGI Home
Prof. Alparslan Acikgence, Vice Rector, Fatih University, Istanbul, Turkey
Luk Bouckaert, Professor Emeritus of Ethics at the Catholic University of Leuven (K.U. Leuven,) and President of the European SPES Forum, Belgium
Dr. Peter Bowman, University Preparatory Certificate for Science and Engineering, University College, London, UK
Rev. Dr. Marcus Braybrooke, President, World Congress of Faiths, Oxford, UK
Dr. Michael Britton, Director, "Global Appreciative Culturing", USA; and Human Dignity and Humiliation Studies, Norway
Mr. Sesto G. Castagnoli, Evolutant and Entrepreneur, Founder, WSF World Spirit Forum, Switzerland
Prof. Jamshid Damooei, Co-director, Centre for Leadership and Values, School of Management, California Lutheran University, USA
Prof. Ulrich Duchrow, Professor of Systematic Theology, Heidelberg University, Germany
Prof. Rita Ghesquière, Professor Emeritus of Comparative Literature at the Catholic University of Leuven (KULeuven) Belgium
Dr. Linda Groff, Professor, Political Science and Future Studies, and Coordinator, Behavioral Science Undergraduate Program, California State University, and Director, Global Options and Evolutionary Futures Consulting, USA
Ann Hallock, former Prof. Of Behavioral Medicine, Michigan State University, USA
Mr. Peter Holland, School of Economic Science, London, UK
Prof. Yahya Kamalipour, Head, Department of Communication, Purdue University Calumet, USA
Dr. April Laskey, Professor Emeritus of Economics, Nazareth College, Rochester, New York, USA
Mr. Ian Mason, Principal of the School of Economic Science, London, UK
Prof. Fr. Peter Milward SJ, Professor Emeritus and Director, Renaissance Institute, Sofia University, Japan
Dr. Ana Langovic Milicevic, Professor of Management and Inter-Cultural Management, University of Kragujevac, Belgrade, Serbia
Dr. Angela Mitchell, Associate Professor of Business Administration, Area Coordinator for Accounting, Business Administration, and Economics, Wilmington College, Ohio, USA
Ellen Novar, Associate Professor of Economics and Business Administration, Wilmington College, Ohio, USA
Dr. Fabio Petito, International Relations, School of Global Studies, University of Sussex, UK
Dr. Stephen Potthoff, Associate Professor of Religion and Philosophy, Wilmington College, Ohio, USA
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Dr. Uli Spalthoff, Human Dignity and Humiliation Studies, Germany
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Dr. Shiv Talwar, President, Spiritual Heritage Education Network Inc; Kitchener, Canada
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text:
Siyan's Newman Canonization Pilgrimage
Last week I had the opportunity to make a pilgrimage to England and Rome. The theme was the life of John Henry Newman. He was canonized last Sunday as a Catholic saint along with four other saints. We were a group of five men, Catholics as well as Lutherans interested in Catholicism.
The journey was important to me and my traveling companions in many ways, as Saint John Henry Newman was perhaps the first Protestant convert to Catholicism to be made a saint. St John Henry Newman was originally an Anglican priest who later converted to Catholicism, and was then ordained a Catholic priest. Later still he was created a cardinal. He strongly influenced the return of Catholic faith to Anglican England.
At St Peter's Square for the Canonisation Mass.
Oxford and Birgminham, London and Rome
We first traveled to London, where we stayed in an Opus Dei center. We spent a day in Oxford, a day in Birmingham and a day in London. All the destinations we visited were related to Newman in one way or another. In Oxford we saw St Mary's Church, where he used to preach as an Anglican priest and filled the church hall, and Littlemore, where he was received into the Catholic Church.
A mosaic in the Church of the Multiplication of the Loaves and Fish in the Holy Land.
This is a blog about being young and Catholic in Finland.
Yiran Chen
Yiran, or Maria Micaela. Born in China but baptised in Spain (2015), student in Turku. Participant of the Youth Weekends at Stella Maris. Cooking, running and photography are my hobbies.
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text:
Q: Split a single Excel column into multiple columns in excel I have a data set of five years and each day consists of 24 observations presented in a single column mean first 24 observations of day 1 followed by the next 24 observations of the day2 similarly for all five years, so I want to split this single excel column into 24 columns (meaning that a single row having 24 observations each) represents a single day.
Can anyone help?

A: The title of your question doesn't seem to match the text of the question.
If you have multiple observations, one in each cell, going down a single column you could use INDEX to 'split' the data into rows of 24 columns.
For example, if the data is currently in column A starting at row 1 you could put this formula in B1 and copy it across and down.
=INDEX($A:$A,COLUMN()-1+24*(ROW()-1))

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text:
\subsection{Overview}

The hard instances we construct are based on propositional XOR-formulas that can also be viewed as systems of linear equations over the $2$-element field $\mathbb F_2$.

Let $V$ be a finite set which we interpret as a set of variables that take values in $\{0,1\}$.
An \emph{XOR-constraint (over $V$)} is a pair $(C,a)$ where $C \subseteq V$ and $a \in \{0,1\}$.
The reader is encouraged to think of such a constraint as the equation $x_1 + \dots + x_k \equiv a \bmod 2$ where $C = \{x_1,\dots,x_k\}$ is the set of those variables that appear on the left side of the equation.
Let $\mathcal C$ be a set of XOR-constraints.
We define the \emph{arity} of $\mathcal C$ to be the maximum cardinality of $C$ for any pair $(C,a) \in \mathcal C$.

We can translate a set of XOR-constraints into a pair of relational structures as follows.
Let $\mathcal C$ be a set of XOR-constraints over a set $V$.
Also suppose that $V = \{x_1,\dots,x_n\}$.
We define $\mathfrak A = \mathfrak A(\mathcal C)$ and $\mathfrak B = \mathfrak B(\mathcal C)$ as follows.
We set $V(\mathfrak A) = V(\mathfrak B) \coloneqq V \times \{0,1\}$, i.e., each element of the structures $\mathfrak A$ and $\mathfrak B$ corresponds to an assignment of a single variable.
For each $i \in [n]$, we add a unary relation $X_i$ and set $X_i^{\mathfrak A} = X_i^{\mathfrak B} \coloneqq \{(x_i,0),(x_i,1)\}$.
Finally, for every constraint $(C,a) \in \mathcal C$ with $C = \{x_{i_1},\dots,x_{i_k}\}$ we introduce a $k$-ary relation $R_{C,a}$ and define
\[R_{C,a}^{\mathfrak A} \coloneqq \left\{\Big((x_{i_1},b_1),\dots,(x_{i_k},b_k)\Big) \;\middle|\; b_1,\dots,b_k \in \{0,1\}, \sum_{j = 1}^{k} b_j \equiv 0 \mod 2\right\}\]
and
\[R_{C,a}^{\mathfrak B} \coloneqq \left\{\Big((x_{i_1},b_1),\dots,(x_{i_k},b_k)\Big) \;\middle|\; b_1,\dots,b_k \in \{0,1\}, \sum_{j = 1}^{k} b_j \equiv a \mod 2\right\}\]

Instead of analysing the Weisfeiler-Leman algorithm directly on $\mathfrak A(\mathcal C)$ and $\mathfrak B(\mathcal C)$, it turns out to more convenient to consider the following game that is directly played on $\mathcal C$ and is known to capture the same information as applying the Weisfeiler-Leman algorithm to the associated structures.

Let $\mathcal C$ be a set of XOR-constraints over a set $V$.
Let $k \in \mathbb{N}$ such that $\mathcal C$ has arity at most $k$.
A partial assignment $\beta \colon X \rightarrow \{0,1\}$ with $X \subseteq V$ \emph{violates} an XOR-constraint $(C,a) \in \mathcal C$ if $C \subseteq X$ and
\begin{equation}
 \sum_{x \in C} \beta(x) \not\equiv a \mod 2.
\end{equation}
For a partial assignment $\beta_0 \colon X_0 \rightarrow \{0,1\}$ with $|X_0| \leq k$ the \emph{$r$-round $k$-pebble game} $\mathcal G_k^{r}(V,\mathcal C,\beta_0)$ is played as follows:
\begin{itemize}
 \item The game has two players called Verifier and Falsifier.
 \item The game is played in rounds with initial position $\beta_0$.
 \item Suppose $\beta\colon X \rightarrow \{0,1\}$ is the current position. Then the next round consists of the following steps:
  \begin{itemize}
   \item Falsifier chooses $x \in V \setminus X$ and $X' \subseteq X$ such that $|X' \cup \{x\}| \leq k$.
   \item Verifier chooses $b \in \{0,1\}$.
   \item The game moves to position $\beta'\colon X' \cup \{x\} \rightarrow \{0,1\}$ with $\beta'(x') = \beta(x')$ for $x' \in X'$ and $\beta'(x) = b$.
  \end{itemize}
 \item Falsifier wins a play if within the first $r$ rounds an assignment $\beta$ violates some XOR-constraint $(C,a) \in \mathcal C$ (if $r = 0$, then Falsifier wins if the initial assignment $\beta_0$ violates some constraint in $\mathcal C$).
 \item Verifier wins a play if Falsifier does not win within the first $r$ rounds.
\end{itemize}

We say Falsifier (respectively Verifier) wins the game $\mathcal G_k^{r}(V,\mathcal C,\beta_0)$ if Falsifier (respectively Verifier) has a winning strategy for the game.
The \emph{$k$-pebble game} $\mathcal G_k(V,\mathcal C,\beta_0)$ is played in the same way, but without any restriction on the number of rounds played.

The following lemma relates the pebble game $\mathcal G_{k}^{r}(V,\mathcal C,\emptyset)$ to bounded-variable fragments of first-order logic and thereby, using Corollary \ref{cor:wl-logic-distinguish-structures}, also to the Weisfeiler-Leman algorithm.
(Here, we use $\emptyset$ to denote the empty assignment, i.e., the domain $X_0$ of the initial partial assignment $\beta_0$ is empty.)

\begin{lemma}[{\cite[Lemma 2.1]{BerkholzN16}}]
 \label{la:translate-bounds}
 Let $k,r \in \mathbb N$ such that $r > 0$ and $k \geq 3$.
 Let $\mathcal C$ be a set of XOR-constraints over a universe $V$ of arity at most $k$.
 Then the following statements are equivalent:
 \begin{enumerate}[label = (\roman*)]
  \item Falsifier wins the $r$-round $k$-pebble game $\mathcal G_{k}^{r}(V,\mathcal C,\emptyset)$.
  \item There exists a sentence $\varphi \in \LLkq{k}{r}$ such that $\varphi \models \mathfrak A(\mathcal C)$ and $\varphi \not\models \mathfrak B(\mathcal C)$.
  \item There exists a sentence $\varphi \in \LCkq{k}{r}$ such that $\varphi \models \mathfrak A(\mathcal C)$ and $\varphi \not\models \mathfrak B(\mathcal C)$.
 \end{enumerate}
\end{lemma}

To obtain a set of XOR-constraints on which Falsifier requires a large number of rounds to win the pebble game, we proceed in two steps.
First, for every sufficiently large $\ell_{\sf hi} \geq \ell_{\sf lo}$, we construct a set of XOR-constraints such that Falsifier wins the $\ell_{\sf lo}$-pebble game, but still requires a linear number of rounds to win the $\ell_{\sf hi}$-pebble game.
This is formalized by the next lemma which forms the main technical contribution of this section.

\begin{lemma}
 \label{la:many-rounds}
 There are absolute constants $\ell_{\sf lo} \geq 2$ and $\delta > 1$ such that for every $\ell_{\sf hi} \geq \ell_{\sf lo}$ and every $r \geq 1$
 there is a set of XOR-constraints $\mathcal C$ of arity at most $\ell_{\sf lo}$ over a set $V$ of size $|V| \leq \delta \cdot \ell_{\sf hi}^{2} \cdot r$ such that Falsifier
 \begin{enumerate}[label = (\alph*)]
  \item wins the $\ell_{\sf lo}$-pebble game $\mathcal G_{\ell_{\sf lo}}(V,\mathcal C,\emptyset)$, but
  \item does not win the $r$-round $\ell_{\sf hi}$-pebble game $\mathcal G_{\ell_{\sf hi}}^{r}(V,\mathcal C,\emptyset)$.
 \end{enumerate}
\end{lemma}

We remark that a similar result has also been obtained in \cite{BerkholzN16}, but with weaker guarantees on the number of rounds required to win the $\ell_{\sf hi}$-pebble game.
It is exactly this improvement that allows us to obtain stronger lower bounds on the iteration number of $k$-WL in comparison to \cite{BerkholzN16}.

Afterwards, we rely on the following hardness compression lemma that reduces the number of variables while essentially maintaining the number of rounds that Falsifier requires to win the game.

\begin{lemma}[Berkholz, Nordstr{\"{o}}m {\cite[Lemma 3.3]{BerkholzN16}}]
 \label{la:condensation}
 There is an absolute constant $\Delta_0 \geq 1$ such that the following holds.
 Suppose $\mathcal C$ is a set of XOR-constraints of arity at most $p$ over a set $V$ of size $|V| = m$.
 Also assume there are parameters $\ell_{\sf lo} > 0$, $\ell_{\sf hi} \geq \Delta_0 \ell_{\sf lo}$ and $r > 0$ such that Falsifier
 \begin{enumerate}[label = (\alph*)]
  \item wins the $\ell_{\sf lo}$-pebble game $\mathcal G_{\ell_{\sf lo}}(V,\mathcal C,\emptyset)$, but
  \item does not win the $r$-round $\ell_{\sf hi}$-pebble game $\mathcal G_{\ell_{\sf hi}}^{r}(V,\mathcal C,\emptyset)$.
 \end{enumerate}
 Let $\Delta$ be an integer such that $\Delta_0 \leq \Delta \leq \ell_{\sf hi}/\ell_{\sf lo}$ and $(2\ell_{\sf hi}\Delta)^{2\Delta} \leq m$.
 Then there is a set of XOR-constraints $\mathcal D$ of arity at most $\Delta p$ over a set $W$ of size $|W| = \lceil m^{3/\Delta}\rceil$ such that Falsifier
 \begin{enumerate}[label = (\Alph*)]
  \item wins the $(\Delta \ell_{\sf lo})$-pebble game $\mathcal G_{\Delta \ell_{\sf lo}}(W,\mathcal D,\emptyset)$, but
  \item does not win the $\frac{r}{2\ell_{\sf hi}}$-round $\ell_{\sf hi}$-pebble game $\mathcal G_{\ell_{\sf hi}}^{r/(2\ell_{\sf hi})}(W,\mathcal D,\emptyset)$.
 \end{enumerate}
\end{lemma}

Combining Lemmas \ref{la:many-rounds} and \ref{la:condensation}, we obtain the following corollary.

\begin{corollary}
 \label{cor:maximal-rounds}
 There are absolute constants $k_0 \in \mathbb N$ and $\alpha,\varepsilon > 0$ such that for every $d \geq k \geq k_0$ and every $n \geq \alpha \cdot d^8 \cdot k^6$
 there is a set of XOR-constraints $\mathcal C$ of arity at most $k$ over a set $V$ of size $|V| \leq n$ such that Falsifier wins the $k$-pebble game $\mathcal G_{k}(V,\mathcal C,\emptyset)$, but does not win the $r$-round $d$-pebble game $\mathcal G_{d}^{r}(V,\mathcal C,\emptyset)$ for all $r \leq n^{\varepsilon k}$.
\end{corollary}

\begin{proof}
 Let $\ell_{\sf lo} \geq 2$ and $\delta > 1$ denote the constants from Lemma \ref{la:many-rounds}.
 Also, let $\Delta_0$ denote the constant from Lemma \ref{la:condensation} and suppose without loss of generality that $\delta, \Delta_0$ are integers and $\Delta_0 \geq 3$.
 We choose
 \[k_0 \coloneqq \max\{\Delta_0\ell_{\sf lo}, 6\ell_{\sf lo}\}.\]

 Let $d \geq k \geq k_0$.
 We set $p \coloneqq \ell_{\sf lo}$, $\ell_{\sf hi} \coloneqq d$ and $\Delta \coloneqq \lfloor\frac{k}{\ell_{\sf lo}}\rfloor$.
 We have $\ell_{\sf hi} = d \geq k \geq k_0 \geq \Delta_0\ell_{\sf lo}$ and $\Delta_0 \leq \frac{k}{\ell_{\sf lo}}$.
 Since $\Delta_0$ is an integer, we conclude that $\Delta_0 \leq \Delta$.

 We define
 \[n_0 \coloneqq \max\{\left(\delta \cdot \ell_{\sf hi}^{2} \cdot (2\ell_{\sf hi}\Delta)^{2\Delta}\right)^{3/\Delta},4 \cdot \delta \cdot \ell_{\sf hi}^{3}\}\]
 and set $\alpha \coloneqq \max\{64\cdot \delta \cdot \ell_{\sf lo}^{-6},4\cdot\delta\}$.
 Let $n \geq \alpha \cdot d^8 \cdot k^6$.
 Using $\Delta \geq \Delta_0 \geq 3$, we get that
 \[\left(\delta \cdot \ell_{\sf hi}^{2} \cdot (2\ell_{\sf hi}\Delta)^{2\Delta}\right)^{3/\Delta} \leq \delta \cdot \ell_{\sf hi}^{2} \cdot (2\ell_{\sf hi}\Delta)^6 \leq 64 \cdot \delta \cdot \ell_{\sf hi}^8 \cdot \left(\frac{k}{\ell_{\sf lo}}\right)^6 \leq \alpha \cdot d^8 \cdot k^6\]
 and $4 \cdot \delta \cdot \ell_{\sf hi}^{3} \leq \alpha \cdot d^3$.
 So in particular $n \geq n_0$.
 Let $r$ be the maximal integer such that
 \begin{equation}
  \label{eq:choose-r}
  \left(\delta \cdot \ell_{\sf hi}^{2} \cdot r\right)^{3/\Delta} \leq n.
 \end{equation}
 Note that $r \geq (2\ell_{\sf hi}\Delta)^{2\Delta}$ since $n \geq n_0$.
 Let $\mathcal C$ be the set of XOR-constraints of arity at most $\ell_{\sf lo}$ over a set $V$ of size $|V| \leq \delta \cdot \ell_{\sf hi}^{2} \cdot r$ obtained from Lemma \ref{la:many-rounds}.
 By adding dummy variables, we may assume without loss of generality that $m \coloneqq |V| \geq r \geq (2\ell_{\sf hi}\Delta)^{2\Delta}$.

 By applying Lemma \ref{la:condensation}, we obtain a set of XOR-constraints $\mathcal D$ of arity at most $\Delta p$ over a set $W$ of size $|W| = \lceil m^{3/\Delta}\rceil$ such that Falsifier
 \begin{enumerate}[label = (\Alph*)]
  \item wins the $(\Delta \ell_{\sf lo})$-pebble game $\mathcal G_{\Delta \ell_{\sf lo}}(W,\mathcal D,\emptyset)$, but
  \item does not win the $\frac{r}{2\ell_{\sf hi}}$-round $\ell_{\sf hi}$-pebble game $\mathcal G_{\ell_{\sf hi}}^{r/(2\ell_{\sf hi})}(W,\mathcal D,\emptyset)$.
 \end{enumerate}
 First observe that $\Delta p = \lfloor\frac{k}{\ell_{\sf lo}}\rfloor \cdot \ell_{\sf lo} \leq k$ and
 \[|W| = \left\lceil m^{3/\Delta}\right\rceil \leq \left\lceil\left(\delta \cdot \ell_{\sf hi}^{2} \cdot r\right)^{3/\Delta}\right\rceil \leq n.\]
 Since $\Delta \ell_{\sf lo} \leq k$, it holds that Falsifier wins the $k$-pebble game $\mathcal G_{k}(W,\mathcal D,\emptyset)$.
 Moreover, Falsifier does not win the $\frac{r}{2\ell_{\sf hi}}$-round $d$-pebble game $\mathcal G_{d}^{r/(2\ell_{\sf hi})}(W,\mathcal D,\emptyset)$.
 We have that
 \[\left(\delta \cdot \ell_{\sf hi}^{2} \cdot 2r\right)^{3/\Delta} \geq n\]
 since $r$ is the maximal integer to satisfy Equation \eqref{eq:choose-r}.
 This implies that
 \[\frac{r}{2\ell_{\sf hi}} \geq \frac{n^{\Delta/3}}{4 \cdot \delta \cdot \ell_{\sf hi}^{3}} \geq n^{\frac{1}{3}\lfloor\frac{k}{\ell_{\sf lo}}\rfloor - 1} \geq n^{\varepsilon k}\]
 for some sufficiently small absolute constant $\varepsilon > 0$.
\end{proof}

With Corollary \ref{cor:maximal-rounds} at hand, we are now ready to prove Theorems \ref{thm:wl-round-lower-bound} and \ref{thm:quantifier-rank-lower-bound}.

\begin{proof}[Proof of Theorem \ref{thm:wl-round-lower-bound}]
 Let $k_0' \in \mathbb N$ and $\alpha',\varepsilon' > 0$ denote the absolute constants from Corollary \ref{cor:maximal-rounds}.

 Let $k_0 \coloneqq \max\{k_0',3\}$.
 We set $\alpha \geq 1$ and $\varepsilon > 0$ in such a way that for all $d \geq k \geq k_0$ and $n \geq \alpha d^8 k^6$ it holds that
 \[\left\lfloor\frac{n}{2}\right\rfloor \geq \alpha'(d+1)^8(k+1)^6\]
 and
 \[\left(\frac{n}{2} - 1\right)^{\varepsilon' k} - d \geq n^{\varepsilon k}.\]
 Now, let us fix some $d \geq k \geq k_0$ and $n \geq \alpha d^8 k^6$.
 Let $d' \coloneqq d+1$, $k' \coloneqq k$ and $n' \coloneqq \lfloor\frac{n}{2}\rfloor$.
 We apply Corollary \ref{cor:maximal-rounds} with parameters $d',k',n'$ and obtain a set of XOR-constraints $\mathcal C$ of arity at most $k'$ over a set $V'$ of size $|V'| \leq n'$ such that Falsifier wins the $k'$-pebble game $\mathcal G_{k'}(V',\mathcal C,\emptyset)$, but does not win the $r'$-round $d'$-pebble game $\mathcal G_{d'}^{r'}(V',\mathcal C,\emptyset)$ for $r' = (n')^{\varepsilon' k'}$.

 Let $\mathfrak A \coloneqq \mathfrak A(\mathcal C)$ and $\mathfrak B \coloneqq \mathfrak B(\mathcal C)$.
 Then $|V(\mathfrak A)| = |V(\mathfrak B)| = 2|V'| \leq 2n' \leq n$.
 Note that we can easily increase the size of both structures by adding isolated elements that do not participate in any relations.
 Also, note that both structures have arity at most $k' = k$.

 By Lemma \ref{la:translate-bounds} and Corollary \ref{cor:wl-logic-distinguish-structures}, $k$-WL distinguishes between $\mathfrak A$ and $\mathfrak B$.
 On the other hand, again by Lemma \ref{la:translate-bounds} and Corollary \ref{cor:wl-logic-distinguish-structures}, $d$-WL does not distinguish $\mathfrak A$ and $\mathfrak B$ after $r \coloneqq r' - d$ refinement rounds.
 We get that
 \[r = r' - d = (n')^{\varepsilon' k'} - d \geq \left(\frac{n}{2} - 1\right)^{\varepsilon' k} - d \geq n^{\varepsilon k}.\qedhere\]
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:quantifier-rank-lower-bound}]
 This follows directly from Corollary \ref{cor:maximal-rounds} and Lemma \ref{la:translate-bounds}.
\end{proof}

The remainder of this section is devoted to the proof of Lemma \ref{la:many-rounds}.

\subsection{The Closure of the Constraint Set}
\label{sec:closure-operator}

The critical step in the proof of Lemma \ref{la:many-rounds} is to argue that Verifier survives a linear number of rounds even for a large number of pebbles.
Here, we rely on an alternative description of winning positions in terms of a closure operator.

Let $k \in \mathbb{N}$.
Let $V$ be a finite set and let $\mathcal C$ be a set of XOR-constraints over $V$ of arity at most $k$.
We define the \emph{$k$-attractor}
\[\attr_k(\mathcal C) \coloneqq \mathcal C \cup \Big\{(C_1 \oplus C_2,a_1 + a_2 \bmod 2) \;\Big|\; (C_1,a_1),(C_2,a_2) \in \mathcal C, |C_1 \oplus C_2| \leq k\Big\}.\]
Here, $C_1 \oplus C_2$ denotes the symmetric difference between the two sets, that is, $C_1 \oplus C_2 \coloneqq (C_1 \cup C_2) \setminus (C_1 \cap C_2)$.

Intuitively speaking, if $x_1 + \dots + x_\ell \equiv a_1 \bmod 2$ and $y_1 + \dots + y_m \equiv a_2 \bmod 2$ are two constraints in $\mathcal C$,
then every satisfying assignment also has to satisfy the equation $x_1 + \dots + x_\ell + y_1 + \dots + y_m \equiv a_1 + a_2 \bmod 2$.
Since all variables appearing in both sets $\{x_1,\dots,x_k\}$ and $\{y_1,\dots,y_m\}$ cancel over $\mathbb F_2$, we only need to keep those variables appearing in the symmetric difference.
In the case that the resulting number of variables is bounded by $k$, we add the corresponding equation to the $k$-attractor of the constraint set.

We define $\cl_k^{(0)}(\mathcal C) \coloneqq \mathcal C$ and $\cl_k^{(r+1)}(\mathcal C) \coloneqq \attr_k(\cl_k^{(r)}(\mathcal C))$ for all $r \geq 0$.
Finally, we define the \emph{$k$-closure} of $\mathcal C$ to be the set $\cl_k(\mathcal C) \coloneqq \cl_k^{(r)}(\mathcal C)$ for the minimal $r \geq 0$ such that $\cl_k^{(r+1)}(\mathcal C) = \cl_k^{(r)}(\mathcal C)$.

The following lemma provides the key method to prove that Verifier can survive a certain number of rounds.

\begin{lemma}
 \label{la:dupl-closure-to-consistency}
 Let $\beta\colon X \rightarrow \{0,1\}$ be a partial assignment with $|X| \leq k$ such that $\beta$ violates no XOR-constraint $(C,a) \in \cl_k^{(r)}(\mathcal C)$.
 Then Verifier wins $\mathcal G_k^{r}(V,\mathcal C,\beta)$.
\end{lemma}

\begin{proof}
 We prove the statement by induction on $r$.
 For $r = 0$ the statement is trivial.
 So suppose $r \geq 1$ and Spoiler chooses $x \in V \setminus X$ and $X' \subseteq X$ such that $|X' \cup \{x\}| \leq k$ in the first round.
 For $b \in \{0,1\}$ let $\beta_b\colon X' \cup \{x\} \rightarrow \{0,1\}$ be the partial assignment with $\beta_b(x') = \beta(x')$ for $x' \in X'$ and $\beta_b(x) = b$.
 Assume towards a contradiction that, for every $b \in \{0,1\}$, there is some XOR-constraint $(C_b,a_b) \in \cl_k^{(r-1)}(\mathcal C)$ violated by $\beta_b$.
 Observe that $x \in C_b$ for both $b \in \{0,1\}$ (since otherwise $\beta$ would violate $(C_b,a_b)$ contradicting our assumption).
 Let $C \coloneqq C_0 \oplus C_1 \subseteq X$ and $a \coloneqq (a_0 + a_1) \bmod 2$.
 Note that $|C| \leq k$ since $C \subseteq X$.
 Then \[\sum_{y \in C} \beta(y) \equiv \sum_{y \in C_0\setminus\{x\}}\beta(y) + \sum_{y \in C_1\setminus\{x\}} \beta(y) \equiv 1 + \sum_{y \in C_0}\beta_0(y) + \sum_{y \in C_1} \beta_1(y) \equiv 1 + a \mod 2\]
 and $(C,a) \in \cl_k^{(r)}(\mathcal C)$.
 Hence, $\beta$ violates some $(C,a) \in \cl_k^{(r)}(\mathcal C)$ which is a contradiction.

 So there is some $b \in \{0,1\}$ such that $\beta_b$ violates no XOR-constraint in $\cl_k^{(r-1)}(\mathcal C)$.
 Verifier chooses such a $b \in \{0,1\}$ and the game moves to position $\beta_b$ which violates no XOR-constraint in $\cl_k^{(r-1)}(\mathcal C)$.
 So Verifier wins $\mathcal G_k^{r-1}(V,\mathcal C,\beta_b)$ by the induction hypothesis which implies that Verifier also wins $\mathcal G_k^{r}(V,\mathcal C,\beta)$.
\end{proof}

\subsection{Layered Graphs and Expansion}

Next, we discuss the construction of certain expander graphs.
Overall, we are aiming to construct what we refer to as \emph{single-neighbor layered expanders}.
Towards this end, we start with constructing standard bipartite expander graphs with an expansion that is close to the minimum degree of one side of the bipartite graph.
We then define \emph{single-neighbor expanders} and observe that bipartite expanders with large expansion also are single-neighbor expanders (with a slightly smaller expansion parameter).
Finally, we obtain \emph{single-neighbor layered expanders} by ``stacking single-neighbor expanders on top of each other''.

\subsubsection{Expander Graphs}

We start by defining standard bipartite expander graphs.

\begin{definition}
 Let $0 < \gamma < 1$ and $\alpha > 1$ be constants and let $G = (V,W,E)$ be a bipartite graph.
 We say that $G$ is an \emph{$(\alpha,\gamma)$-expander} if for every $\emptyset \neq Y \subseteq W$ with $|Y| \leq \gamma |W|$ it holds that \[N(Y) \geq \alpha |Y|.\]
\end{definition}

For more information on expander graphs we refer to \cite{MotwaniR95,Vadhan12}.
The references also contain variants of the following standard argument that guarantees the existence of graphs with good expansion properties.
For our purposes, the crucial property in the lemma below is that the expansion $\alpha$ is relatively close to the degree of the vertices in $W$.

\begin{lemma}
 \label{la:expander}
 There is some number $R_0 \geq 2$ such that for every $r \geq R_0$ and every $n \geq 4r$ there is a $(\frac{3}{4}r,\frac{1}{20r})$-expander $G = (V,W,E)$ such that $|V| = |W| = n$ and $\deg(w) = r$ for all $w \in W$.
\end{lemma}

\begin{proof}
 Suppose $r$ is sufficiently large.
 Let $V,W$ be two sets with $|V| = |W| \geq 4r$.
 We construct a bipartite graph $G = (V,W,E)$ using the following random process:
 for each $w \in W$ we select independently and uniformly at random a set of $r$ distinct neighbors from $V$.
 We prove that, for $r$ sufficiently large, with positive probability the graph $G$ is a $(\frac{3}{4}r,\frac{1}{20r})$-expander.

 Let $n \coloneqq |V| = |W|$. For $X \subseteq V$ and $Y \subseteq W$ let~$p_{X,Y}$ denote the probability that $N(Y) \subseteq X$.
 Then
 \[p_{X,Y} \leq \left(\frac{|X|}{n}\right)^{r\cdot |Y|}.\]
 Furthermore, let $\alpha \coloneqq \frac{3}{4}r$ and $\gamma \coloneqq \frac{1}{20 r}$.
 Let $p$ be the probability that $G$ is not a $(\gamma,\alpha)$-expander.
 Then, using the inequality~$\binom{n}{k}\leq (ne/k)^k$, we get
 \begin{align*}
  p &\leq \sum_{\substack{Y \subseteq W\\|Y| \leq \gamma\cdot n}}\;\sum_{\substack{X \subseteq V\\|X| = \lfloor \alpha|Y|\rfloor}} p_{X,Y}\\
    &\leq \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \sum_{\substack{Y \subseteq W\\|Y| =s}}\;\sum_{\substack{X \subseteq V\\|X| =  \lfloor\alpha|Y|\rfloor}} \left(\frac{|X|}{n}\right)^{r\cdot |Y|}\\
    &\leq \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \binom{n}{s}\binom{n}{\lfloor \alpha s \rfloor} \left(\frac{\alpha s}{n}\right)^{r\cdot s}\\
    &\leq \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \left(\frac{ne}{s}\right)^{s} \left(\frac{ne}{\alpha s}\right)^{\alpha \cdot s} \left(\frac{\alpha s}{n}\right)^{r\cdot s}\\
    &= \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \left[\left(\frac{ne}{s}\right) \left(\frac{ne}{\alpha s}\right)^{\alpha} \left(\frac{\alpha s}{n}\right)^{r}\right]^{s}
    \\
    &= \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \left[\left(\frac{s}{n}\right)^{r-\alpha-1} e^{1+\alpha} \alpha^{r-\alpha}\right]^{s} \\
    &= \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \left[\left(\frac{s}{n}\right)^{r/4-1} e^{1+3r/4} (3r/4)^{r/4}\right]^{s}\\
    &\leq \sum_{s = 1}^{\lfloor \gamma\cdot n\rfloor} \left[\gamma^{r/4-1} e^{1+3r/4} (3r/4)^{r/4}\right]^{s}\;.
 \end{align*}
 Now let $x := \gamma^{r/4-1} e^{1+3r/4} (3r/4)^{r/4}$.
 For~$r$ sufficiently large we get \[x = (20r)^{1-r/4} e^{1+3r/4} (3r/4)^{r/4} = 20er \left(\frac{3e^{3}}{80}\right)^{r/4} <1/10.\]
 It follows that \[p \leq \sum_{s = 1}^{\infty} x^{s} = \frac{x}{1-x} \leq \frac{1}{9}.\]
 In particular, $p < 1$ which implies the existence of the desired expander graph.
\end{proof}

Next, we turn to what we call single-neighbor expanders where each sufficiently small set $Y \subseteq V$ is required to have a large number of neighbors that additionally satisfy the property that they are the neighbor of only a single vertex from $Y$.
Let $G = (V,W,E)$ be a bipartite graph.
For $Y \subseteq W$ we define
\[N^{*}(Y) = \{v \in N(Y) \mid |N(v) \cap Y| = 1\}.\]

\begin{definition}
 Let $0 < \gamma < 1$ and $\alpha > 1$ be constants and let $G = (V,W,E)$ be a bipartite graph.
 We say that $G$ is an \emph{$(\alpha,\gamma)$-single-neighbor expander} if for every $\emptyset \neq Y \subseteq W$ with $|Y| \leq \gamma |W|$ it holds that \[N^{*}(Y) \geq \alpha |Y|.\]
\end{definition}

We can obtain single-neighbor expanders from Lemma \ref{la:expander} by allowing some loss on the expansion parameter $\alpha$.

\begin{corollary}
 \label{cor:single-neighbor-expander}
 There is some number $R_0 \geq 5$ such that for every $r \geq R_0$ and every $n \geq 4r$ there is a $(\frac{1}{4}r,\frac{1}{20r})$-single-neighbor expander $G = (V,W,E)$ such that $|V| = |W| = n$ and $\deg(w) = r$ for all $w \in W$.
\end{corollary}

\begin{proof}
 Choose $R_0 \coloneqq \max(5,R_0')$ where $R_0'$ is the constant from Lemma \ref{la:expander} and suppose $r \geq R_0$ and $n \geq 4r$.
 By Lemma \ref{la:expander}, there is a $(\frac{3}{4}r,\frac{1}{20r})$-expander $G = (V,W,E)$ such that $|V| = |W| = n$ and $\deg(w) = r$ for all $w \in W$.
 We claim that $G$ is a $(\frac{1}{4}r,\frac{1}{20r})$-single-neighbor expander.
 Let $Y \subseteq W$ with $|Y| \leq \frac{n}{20r}$.
 Then $|N(Y)| \geq \frac{3}{4}r|Y|$.
 Furthermore $|N(Y)| = |N^{*}(Y)| + |\{v \in N(Y) \mid |N(v) \cap Y| \geq 2\}| \leq |N^{*}(Y)| + \frac{1}{2}r|Y|$ because every vertex in $Y$ has degree $r$.
 Thus, $|N^{*}(Y)| \geq \frac{1}{4}r|Y|$.
\end{proof}

\subsubsection{Layered Graphs}

Now, we turn to the construction of single-neighbor layered expanders which is the main tool for constructing the desired constraint sets in the proof of Lemma \ref{la:many-rounds}.
We start by defining a certain notion of layered graphs (see also Figure \ref{fig:layered-graph}).

Let $\ell,m \in \mathbb N$.
An \emph{$(\ell \times m)$-layered graph} is a bipartite graph $G = (V,W,E)$ for which there are partitions $V = V_0 \uplus \dots \uplus V_\ell$ and $W = W_1 \uplus \dots \uplus W_\ell$ such that
\begin{enumerate}
 \item $|V_i| = m$ for all $i \in [0,\ell]$,
 \item $|W_i| = m$ for all $i \in [\ell]$,
 \item $N_G(W_i) \subseteq V_{i-1} \cup V_i$ for all $i \in [\ell]$, and
 \item $G[V_i \cup W_i]$ is $1$-regular (i.e., a matching) for all $i \in [\ell]$.
\end{enumerate}

\begin{figure}
 \centering
 \begin{tikzpicture}
  \draw[thick,gray] (0,0) ellipse (4cm and 0.3cm);
  \draw[thick,gray] (0,1) ellipse (4cm and 0.3cm);
  \draw[thick,gray] (0,2) ellipse (4cm and 0.3cm);
  \draw[thick,gray] (0,3) ellipse (4cm and 0.3cm);
  \draw[thick,gray] (0,5) ellipse (4cm and 0.3cm);
  \draw[thick,gray] (0,6) ellipse (4cm and 0.3cm);

  \node at (5,0) {$V_0$};
  \node at (5,1) {$W_1$};
  \node at (5,2) {$V_1$};
  \node at (5,3) {$W_2$};
  \node at (5,5) {$W_\ell$};
  \node at (5,6) {$V_\ell$};

  \node at (-2,4) {$\vdots$};
  \node at (2,4) {$\vdots$};

  \foreach \i in {0,1,2,3,5,6}{
   \foreach \j in {0,1,2,3,4,8,9,10}{
    \node[smallvertex] (v-\i-\j) at (-3 + \j*0.6,\i) {};
   }
   \node at (0.6,\i) {$\dots$};
  }

  \foreach \i/\j in {1/2,5/6}{
   \foreach \k in {0,1,2,3,4,8,9,10}{
    \draw (v-\i-\k) edge (v-\j-\k);
   }
  }

  \foreach \i/\j in {0/1,2/3}{
   \foreach \k/\m in {0/0,0/2,1/0,1/1,1/2,2/1,2/3,3/2,3/3,3/4,4/3,4/4,8/8,8/9,9/8,9/10,10/9,10/10}{
    \draw (v-\j-\k) edge (v-\i-\m);
   }
  }

 \end{tikzpicture}
 \caption{Visualization of $(\ell \times m)$-layered graphs.}
 \label{fig:layered-graph}
\end{figure}

With this, we are now ready to define the notion of single-neighbor layered expanders.

\begin{definition}
 Let $\ell,m \geq 2$.
 Let $0 < \gamma < 1$ and $\alpha > 1$ be constants and let $G = (V,W,E)$ be an $(\ell \times m)$-layered graph.
 We say that $G$ is an \emph{$(\alpha,\gamma)$-single-neighbor $(\ell \times m)$-layered expander} if for every $\emptyset \neq Y \subseteq W$ with $|Y| \leq \gamma m$ it holds that \[N^{*}(Y) \geq \alpha |Y|.\]
\end{definition}

Note that an $(\alpha,\gamma)$-single-neighbor $(\ell \times m)$-layered expander is not a $(\alpha,\gamma)$-single-neighbor expander since we are only considering sets $Y \subseteq W$ of size $|Y| \leq \gamma m$, i.e., we are only considering sets that are smaller (by a factor of $\gamma$) than a single layer of a layered graph.
In particular, the reader is encouraged to think of $\ell$ being much larger than $m$.
In this case, such a graph is far from being a (global) expander, but the key property is that it behaves like an expander when only considering a few layers of the graph.

By again allowing some small loss on the expansion parameter $\alpha$, we can obtain single-neighbor layered expanders by ``stacking $\ell$ copies of a single-neighbor expander on top of each other''.

\begin{corollary}
 \label{cor:single-neighbor-layered-expander}
 There is some number $R_0 \geq 9$ such that for every $r \geq R_0$, every $\ell \geq 1$, and every $m \geq 4r$ there is a $(\frac{1}{4}r - 1,\frac{1}{20r})$-single-neighbor $(\ell \times m)$-layered expander $G = (V,W,E)$ with partitions $V = V_0 \uplus \dots \uplus V_\ell$ and $W = W_1 \uplus \dots \uplus W_\ell$ such that $N_G(w) \cap V_{i-1} = r$ for all $w \in W_i$ and all $i \in [\ell]$.
\end{corollary}

\begin{proof}
 Choose $R_0 \coloneqq \max(9,R_0')$ where $R_0'$ is the constant from Corollary \ref{cor:single-neighbor-expander} and suppose $r \geq R_0$, $\ell \geq 1$, and $m \geq 4r$.
 By Corollary \ref{cor:single-neighbor-expander}, there is a $(\frac{1}{4}r,\frac{1}{20r})$-single-neighbor expander $G' = (V',W',E')$ such that $|V'| = |W'| = m$ and $\deg_{G'}(w') = r$ for all $w' \in W'$.
 Suppose $V' = \{v_1',\dots,v_m'\}$ and $W' = \{w_1',\dots,w_m'\}$.

 We set $V_i \coloneqq \{v_{i,1},\dots,v_{i,m}\}$ for all $i \in [0,\ell]$ and $W_i \coloneqq \{w_{i,1},\dots,w_{i,m}\}$ for all $i \in [\ell]$.
 Also, we set
 \[E \coloneqq \{v_{i-1,j}w_{i,k} \mid i \in [\ell], v_j'w_k' \in E'\} \cup \{v_{i,j}w_{i,j} \mid i \in [\ell], j \in [m]\}.\]
 Clearly, $G = (V,W,E)$ is an $(\ell \times m)$-layered graph.

 Let $\alpha \coloneqq \frac{1}{4}r$ and $\gamma \coloneqq \frac{1}{20r}$.
 Also let $Y \subseteq W$ such that $|Y| \leq \gamma m$.
 We define $Y_i \coloneqq Y \cap W_i$ for all $i \in [\ell]$.
 Observe that $|Y_i| \leq \gamma m$ for all $i \in [\ell]$ and $Y_1,\dots,Y_\ell$ forms a partition of $Y$.
 Let $\mathcal{I} \coloneqq \{i \in [\ell] \mid Y_i \neq \emptyset\}$.
 Since $G'$ is an $(\alpha,\gamma)$-single-neighbor expander, we conclude that
 \[|N^{*}(Y_i) \cap V_{i-1}| \geq \alpha |Y_i|\]
 for all $i \in \mathcal{I}$.
 Moreover, since $G[V_{i} \cup W_{i}]$ is $1$-regular (i.e., a matching) for all $i \in [\ell]$, we conclude that
 \[|N^{*}(Y) \cap V_{i-1}| \geq \alpha |Y_i| - |Y_{i-1}|\]
 for all $i \in \mathcal{I}$ (we set $Y_0 \coloneqq \emptyset$).
 So overall
 \[|N^{*}(Y)| \geq \sum_{i \in \mathcal{I}} \alpha |Y_i| - |Y_{i-1}| \geq \sum_{i \in \mathcal{I}} (\alpha - 1) |Y_i| = (\alpha - 1)|Y|\]
 as desired.
\end{proof}

\subsection{Constraint Sets from Layered Expanders}

Now, we turn to the construction of constraint sets from layered graphs.
For a bipartite graph $G = (V,W,E)$ we define the XOR-constraint set $\mathcal C_G \coloneqq \{(N(w),0) \mid w \in W\}$ over the variable set $V$.
Slightly abusing notation, for $C \subseteq V$, we shall also write $C \in \mathcal C_G$ if $(C,0) \in \mathcal C_G$.

The basic idea for the construction of the XOR-constraint set $\mathcal C$ is to take a layered graph $G = (V,W,E)$ with partitions $V = V_0 \uplus \dots \uplus V_\ell$ and $W = W_1 \uplus \dots \uplus W_\ell$, and set
\[\mathcal C \coloneqq \mathcal C_G \cup \big\{(\{x\},0) \mathrel{\big|} x \in V_0\big\} \cup \big\{(\{x_\ell\},1)\big\}\]
for some arbitrary $x_\ell \in V_\ell$.
It is not difficult to see that this constraint set is unsatisfiable.
Indeed, every variable in layer $V_0$ needs to be set to $0$, and if all variables in layer $V_{i-1}$ are set to $0$, then the constraints obtained from the vertices in $W_i$ enforce that every variable in layer $V_i$ needs to be set to $0$ as well (using that $G[V_i \cup W_i]$ is a matching).
This inductive argument can be easily turned into a winning strategy for Falsifier that requires $O(\ell)$ many rounds (assuming the degree of all vertices in $V$ is bounded by some absolute constant $d \leq k$ where $k$ denotes the number of variables available in the game).

Now, the central claim is that, if we start with a single-neighbor layered expander, this strategy is essentially optimal.
Let us suppose for the moment that only constraints from $\mathcal C_G$ are present and consider the $k$-closure $\cl_k(\mathcal C_G)$.
What we need to avoid is that $\cl_k(\mathcal C_G)$ contains some constraint that is ``non-local''.
For example, if $\cl_k(\mathcal C_G)$ would contain a constraint $(\{x_1,x_2,x_3\},0)$ such that $x_1,x_2 \in V_0$ and $x_3 \in V_\ell$, then Falsifier could use such a (derived) constraint to immediately conclude that certain variables in the last layer need to be set to $0$ and potentially follow a different strategy to win the game faster.
The main point is that, by using single-neighbor layered expanders, we ensure that all ``relevant'' constraints in $\cl_k(\mathcal C_G)$ are ``local'', i.e., they can only contain variables of $O(k)$ consecutive layers.
(Here, the reader may note that if $|N(v_1) \cup N(v_2)| \leq k$ then $(N(v_1) \oplus N(v_2),0)$ is always contained in the closure even if $v_1$ and $v_2$ are far apart. However, in such a case, $N(v_1) \cap N(v_2) = \emptyset$ and the derived constraint $(N(v_1) \cup N(v_2),0)$ is not ``relevant'' since, whenever it is violated by a partial assignment, one of the constraints associated with $v_1$ or $v_2$ is also violated.)
This way, even when adding all constraints from $\cl_k(\mathcal C_G)$ to the initial set, the best that Falsifier can do is essentially to follow the above inductive strategy (with the exception that Falsifier may skip up to $O(k)$ layers in one step which, however, does not cause any problems for our arguments).

For technical reasons, the formal arguments slightly deviate from the intuitive ideas described above.
To start, instead of working with the $k$-closure $\cl_k(\mathcal C_G)$, it turns out to be more convenient to work with the following set.

Let $\alpha > 1$ and $k \geq 1$.
We define the set
\[\cl_{k,\alpha}^{*}(\mathcal C_G) \coloneqq \left\{\left(\bigoplus_{D \in \mathcal D}D,0\right) \;\middle|\; \emptyset \neq \mathcal D \subseteq \mathcal C_G, |\mathcal D| \leq \frac{k}{\alpha}, \Big|\bigoplus_{D \in \mathcal D}D\Big| \leq k\right\}.\]
The next lemma implies that $\cl_k(\mathcal C_G) \subseteq \cl_{k,\alpha}^{*}(\mathcal C_G)$ if $G$ is a suitable single-neighbor layered expander.

\begin{lemma}
 \label{la:single-neighbor-layered-small-sum-closed}
 Suppose $\alpha > 1$ and $0 < \gamma < 1$.
 Let $G = (V,W,E)$ be an $(\alpha,\gamma)$-single-neighbor $(\ell \times m)$-layered expander such that $\deg(w) \leq d$ for all $w \in W$ and suppose $d \leq k \leq \frac{1}{2}\gamma m$.
 Then
 \[\attr_k\left(\cl_{k,\alpha}^{*}(\mathcal C_G)\right) = \cl_{k,\alpha}^{*}(\mathcal C_G).\]
\end{lemma}

\begin{proof}
 Let $\mathcal C^{*} \coloneqq \cl_{k,\alpha}^{*}(\mathcal C_G)$.
 Suppose $C \in \attr_k(\mathcal C^{*})$, that is, there are $C_1,C_2 \in \mathcal C^{*}$ such that $C \coloneqq C_1 \oplus C_2$ and $|C| \leq k$.
 By definition, there are integers $s,t \leq \frac{k}{\alpha}$ and $D_1,\dots,D_s,D_{s+1},\dots,D_{s+t} \in \mathcal C_G$ such that $C_1 = D_1 \oplus \dots \oplus D_s$ and $C_2 = D_{s+1} \oplus \dots \oplus D_{s+t}$.
 Moreover, $D_1,\dots,D_s$ are pairwise distinct as well as $D_{s+1},\dots,D_{s+t}$ are pairwise distinct.
 We have $C = D_1 \oplus \dots \oplus D_{s+t}$.
 Let
 \[\mathcal D \coloneqq \{D_1,\dots,D_s\} \oplus \{D_{s+1},\dots,D_{s+t}\}\] and let $Y \coloneqq \{w \in W \mid N(w) \in \mathcal D\}$.
 Clearly, $C = \bigoplus_{D \in \mathcal D} D$.
 Suppose towards a contradiction that $|\mathcal D| > \frac{k}{\alpha}$.
 Then $|Y| > \frac{k}{\alpha}$ and moreover, $|Y| \leq s+t \leq 2\frac{k}{\alpha} \leq 2k \leq \gamma m$ and thus, $|N^{*}(Y)| \geq \alpha|Y| > k$. But on the other hand $N^{*}(Y) \subseteq C$ which implies that $|N^{*}(Y)| \leq k$.
 This is a contradiction.
 So $|\mathcal D| \leq \frac{k}{\alpha}$ which implies that $C \in \mathcal C^{*}$ as desired.
\end{proof}

\begin{lemma}
 \label{la:single-neighbor-layered-no-singletons}
 Suppose $\alpha > 1$ and $0 < \gamma < 1$.
 Let $G = (V,W,E)$ be an $(\alpha,\gamma)$-single-neighbor $(\ell \times m)$-layered expander such that $\deg(w) \leq d$ for all $w \in W$ and suppose $d \leq k \leq \frac{1}{2}\gamma m$.
 Then $|C| \geq 2$ for all $C \in \cl_{k,\alpha}^{*}(\mathcal C_G)$.
\end{lemma}

\begin{proof}
 Let $C \in \cl_{k,\alpha}^{*}(\mathcal C_G)$ and let $C_1,\dots,C_s \in \mathcal C_G$ such that $C = C_1 \oplus \dots \oplus C_s$ for some $s \leq \frac{k}{\alpha} \leq k$.
 Furthermore, let $Y \coloneqq \{w \in W \mid \exists i \in [s] \colon N(w) = C_i\}$.
 Observe that $1 \leq |Y| \leq k \leq \gamma m$.
 Then $N^{*}(Y) \subseteq C$ and thus, $|C| \geq |N^{*}(Y)| \geq \alpha |Y| > 1$.
\end{proof}

Next, we prove that Falsifier wins the pebble game if we set all variables in layer $V_0$ to $0$, and a single variable in the last layer $V_\ell$ to $1$.
For technical reasons, we do not add $(\{x_\ell\},1)$ to the constraint set, but rather consider an initial assignment that assigns value $1$ to variable $x_\ell$.

\begin{lemma}
 \label{la:falsifier-win}
 Let $G = (V,W,E)$ be an $(\ell \times m)$-layered graph with partitions $V = V_0 \uplus \dots \uplus V_\ell$ and $W = W_1 \uplus \dots \uplus W_\ell$ such that $\deg(w) \leq k$ for all $w \in W$.
 Let $x_\ell \in V_\ell$ and suppose $\beta_\ell\colon\{x_\ell\} \rightarrow \{0,1\}$ is the partial assignment defined via $\beta_\ell(x_\ell) = 1$.
 Then Falsifier wins $\mathcal G_k(W,\mathcal C,\beta_\ell)$ where
 \[\mathcal C \coloneqq \mathcal C_G \cup \big\{(\{x\},0) \mathrel{\big|} x \in V_0\big\}.\]
\end{lemma}

\begin{proof}
 We prove by induction on $i = 0,\dots,\ell$ that Falsifier wins $\mathcal G_k(V,\mathcal C,\beta_i)$ where $\beta_i$ is any partial assignment for which $\beta_i(x_i) = 1$ for some $x_i \in V_i$.

 The base case $i = 0$ is trivial since $(\{x_0\},0) \in \mathcal C$ for every $x_0 \in V_0$.
 For the inductive step, suppose $i \in [\ell]$ and consider some partial assignment $\beta_i$ for which there is some $x_i \in V_i$ such that $\beta_i(x_i) = 1$.
 Since $G = (V,W,E)$ is an $(\ell \times m)$-layered graph, there is a unique vertex $w_i \in W_i$ such that $w_ix_i \in E$.
 Moreover, $N_G(w_i) \subseteq V_{i-1} \cup V_i$.
 If $N_G(w_i) = \{x_i\}$, then $(\{x_i\},0) \in \mathcal C_G$ and Falsifier wins immediately.
 So suppose that $N_G(w_i) \cap V_{i-1} \neq \emptyset$.
 Since $\deg(w_i) \leq k$, Falsifier can move to a partial assignment $\beta_{i-1}\colon X_i \rightarrow \{0,1\}$ where $X_i = N_G(w_i)$ and $\beta_{i-1}(x_i) = 1$.
 If $\beta_{i-1}$ violates the XOR-constraint $(X_i,0)$, then Falsifier wins immediately.
 Otherwise, $\sum_{y \in X_i} \beta_{i-1}(y) = 0$.
 Together with the fact that $\beta_{i-1}(x_i) = 1$, this implies that there is some $x_{i-1} \in X_i \cap V_{i-1}$ such that $\beta_{i-1}(x_{i-1}) = 1$.
 So Falsifier wins by the induction hypothesis.
\end{proof}

The next lemma forms the key technical lemma stating that Falsifier requires a large number of rounds to win if the constraint set is obtained from a single-neighbor layered expander.

\begin{lemma}
 \label{la:verifier-win-few-rounds}
 Suppose $\alpha > 1$ and $0 < \gamma < 1$.
 Let $G = (V,W,E)$ be an $(\alpha,\gamma)$-single-neighbor $(\ell \times m)$-layered expander with partitions $V = V_0 \uplus \dots \uplus V_\ell$ and $W = W_1 \uplus \dots \uplus W_\ell$ such that $\deg(w) \leq d$ for all $w \in W$ and suppose $d \leq k \leq \frac{1}{2}\gamma m$.

 Let $x_\ell \in V_\ell$ and suppose $\beta_\ell\colon\{x_\ell\} \rightarrow \{0,1\}$ is the partial assignment defined via $\beta_\ell(x_\ell) = 1$.
 Then Verifier wins $\mathcal G_k^{r-1}(V,\mathcal C,\beta_\ell)$ where
 \[\mathcal C \coloneqq \mathcal C_G \cup \big\{(\{x\},0) \mathrel{\big|} x \in V_0\big\}\]
 and $r \coloneqq \lfloor\ell/2k\rfloor$.
\end{lemma}

\begin{proof}
 Let
 \[\mathcal C_G^{*} \coloneqq \cl_{k,\alpha}^{*}(\mathcal C_G) \cup \{(\emptyset,0)\}\]
 and define $\mathcal C^{*} \coloneqq \mathcal C_G^{*} \cup \{(\{x\},0) \mid x \in V_0\}$.
 We show that Verifier wins $\mathcal G_k^{r-1}(V,\mathcal C^{*},\beta_\ell)$ which clearly implies the claim since $\mathcal C \subseteq \mathcal C^{*}$.
 By Lemma \ref{la:dupl-closure-to-consistency}, it suffices to show that $\beta_\ell$ violates no XOR-constraint from the set $\cl_k^{(r-1)}(\mathcal C^{*})$, or equivalently $(\{x_\ell\},0) \notin \cl_k^{(r-1)}(\mathcal C^{*})$.

 We define
 \[\mathcal V_i \coloneqq \bigcup_{j = 0}^{2ik} V_j\]
 for all $i \in \{0,\dots,\lfloor\ell/2k\rfloor\}$.
 Finally, we define
 \[\mathcal C_i^{*} \coloneqq \Big\{C \subseteq V \;\Big|\; |C| \leq k, C = D \oplus U \text{ for some } D \in \mathcal C_G^{*}, U \subseteq \mathcal V_i\Big\}\]
 for all $i \in \{0,\dots,\lfloor\ell/2k\rfloor\}$.

 \begin{claim}
  \label{claim:attr-update}
  $\attr_k(\mathcal C_i^{*}) \subseteq \mathcal C_{i+1}^{*}$ for all $i \in \{0,\dots,\lfloor\ell/2k\rfloor-1\}$.
 \end{claim}
 \begin{claimproof}
  Let $C_1,C_2 \in \mathcal C_i^{*}$ such that $|C_1 \oplus C_2| \leq k$.
  Let $C \coloneqq C_1 \oplus C_2$.
  For $j \in \{1,2\}$ pick $D_j \in \mathcal C_G^{*}$ and $U_j \subseteq \mathcal V_i$ such that $C_j = D_j \oplus U_j$.
  Let $U' \coloneqq U_1 \oplus U_2$.
  Clearly, $U' \subseteq \mathcal V_i$ and $C = D_1 \oplus D_2 \oplus U'$.

  Let $Y_j \subseteq W$, $j \in \{1,2\}$, be a set of vertices of size $|Y_j| \leq \frac{k}{\alpha} < k$ such that $D_j = \bigoplus_{w \in Y_j} N(w)$.
  Then there is some $\lambda \in \{2ik+1,\dots,2(i+1)k\}$ such that $W_\lambda \cap (Y_1 \cup Y_2) = \emptyset$.
  We define
  \[Y_j^{< \lambda} \coloneqq Y_j \cap W_{< \lambda}\]
  where $W_{< \lambda} \coloneqq \bigcup_{\mu < \lambda} W_\mu$ and
  \[Y_j^{> \lambda} \coloneqq Y_j \cap W_{> \lambda}\]
  where $W_{> \lambda} \coloneqq \bigcup_{\mu > \lambda} W_\mu$.
  Moreover, let
  \[C_j^{> \lambda} \coloneqq \bigoplus_{w \in Y_j^{> \lambda}}N(w) \subseteq C_j\]
  for both $j \in \{1,2\}$ and
  \[C_{> \lambda} \coloneqq C_1^{> \lambda} \oplus C_2^{> \lambda} \subseteq C.\]
  Hence, $|C_j^{> \lambda}| \leq k$ and $|C_{> \lambda}| \leq k$.
  So $C_j^{> \lambda} \in \mathcal C_G^{*}$ for both $j \in \{1,2\}$.
  It follows that $C_{> \lambda} \in \mathcal C_G^{*}$ by Lemma \ref{la:single-neighbor-layered-small-sum-closed}.

  Now, $C = C_{> \lambda} \oplus U$ for some $U \subseteq V_0 \cup \dots \cup V_{\lambda-1} \subseteq \mathcal V_{i+1}$.
  It follows that $C \in \mathcal C_{i+1}^{*}$.
 \end{claimproof}

 Since $\mathcal C^{*} \subseteq \mathcal C_0^{*}$ it follows by induction that
 \begin{equation}
  \cl_k^{(i)}(\mathcal C^{*}) \subseteq \mathcal C_i^{*}
 \end{equation}
 for all $i \in \{0,\dots,\lfloor\ell/2k\rfloor\}$ using Claim \ref{claim:attr-update}.
 So it only remains the prove the following claim.

 \begin{claim}
  $\{x_\ell\} \notin \mathcal C_{r-1}^{*}$.
 \end{claim}
 \begin{claimproof}
  Let $C \in \mathcal C_{r-1}^{*}$ such that $C \cap V_\ell \neq \emptyset$.
  Also pick $D \in \mathcal C_G^{*}$ and $U \subseteq \mathcal V_{r-1}$ such that $C = D \oplus U$ (which exist by the definition of $\mathcal C_{r-1}^{*}$).
  We have that
  \[U \subseteq \mathcal V_{r-1} = \bigcup_{i = 0}^{2k(r-1)} V_i \subseteq \bigcup_{i = 0}^{\ell - 2k} V_i.\]
  Let $Y \subseteq W$ such that $|Y| \leq \frac{k}{\alpha} < k$ and $D = \bigoplus_{w \in Y} N(w)$.
  Let $\lambda \in [\ell]$ be the maximal number such that $Y \cap W_\lambda = \emptyset$.
  Note that $\lambda > \ell - k$ since $|Y| < k$.
  Now let $D' \coloneqq \bigoplus_{w \in Y \cap W_{> \lambda}} N(w)$ where $W_{> \lambda} \coloneqq \bigcup_{\mu > \lambda} W_\mu$.
  Then $D' = C \cap (V_\lambda \cup \dots \cup V_\ell)$ and hence, $|D'| \leq k$.
  It follows that $D' \in \cl_{k,\alpha}^{*}(\mathcal C_G)$ which implies that $|D'| \geq 2$ using Lemma \ref{la:single-neighbor-layered-no-singletons}.
  So $|C| \geq 2$.
 \end{claimproof}
\end{proof}

Finally, we require one more technical lemma that allows us to add the XOR-constraint $(\{x_\ell\},1)$ to the final constraint set.

\begin{lemma}
 \label{la:strategy-assign-one-variable}
 Let $k \geq 2$ and $r \geq 1$.
 Let $V$ be a finite set and let $\mathcal C$ be a set of XOR-constraints over $V$.
 Let $x_0 \in V$ and define $\beta_0\colon\{x_0\} \rightarrow \{0,1\}$ via $\beta_0(x_0) = 1$.
 If Verifier wins $\mathcal G_k^{r}(V,\mathcal C,\beta_0)$, then Verifier also wins $\mathcal G_{k-1}^{r}(V,\mathcal C \cup \{(\{x_0\},1)\},\emptyset)$.
\end{lemma}

\begin{proof}
 Consider a position $\beta\colon X \rightarrow \{0,1\}$ of the game $\mathcal G_{k-1}^{r}(V,\mathcal C \cup \{(\{x_0\},1)\},\emptyset)$.
 Throughout the game, by following a winning strategy for $\mathcal G_k^{r}(V,\mathcal C,\beta_0)$, Verifier can maintain the following properties after every round $\ell \in [0,r]$:
 \begin{enumerate}[label = (\roman*)]
  \item If $x_0 \in X$, then $\beta(x_0) = 1$, and
  \item Verifier wins the game $\mathcal G_k^{r-\ell}(V,\mathcal C,\beta')$ where $\beta' \colon X \cup \{x_0\} \rightarrow \{0,1\}$ is defined via $\beta'(x) \coloneqq \beta(x)$ for all $x \in X$ and $\beta(x_0) \coloneqq 1$.
 \end{enumerate}
 Observe that the condition is satisfied initially since Verifier wins $\mathcal G_k^{r}(W,\mathcal C,\beta_0)$.
 All positions reached this way clearly satisfy all XOR-constraints in $\mathcal C \cup \{(\{x_0\},1)\}$ which implies that Verifier wins $\mathcal G_{k-1}^{r}(W,\mathcal C \cup \{(\{x_0\},1)\},\emptyset)$.
\end{proof}

With this, we are ready to prove Lemma \ref{la:many-rounds}.

\begin{proof}[Proof of Lemma \ref{la:many-rounds}]
 Let $R_0 \geq 9$ denote the constant from Corollary \ref{cor:single-neighbor-layered-expander} and define $\ell_{\sf lo} \coloneqq R_0 + 1$.
 Let $d \coloneqq R_0$, $\alpha \coloneqq \frac{1}{4}d - 1 > 1$ and $\gamma \coloneqq \frac{1}{20d}$.
 Let $\ell_{\sf hi} \geq \ell_{\sf lo}$ and $r \geq 1$ be given.
 We define $k \coloneqq \ell_{{\sf hi}} + 1$.
 Also, let $m \coloneqq 2 \cdot \frac{k}{\gamma} = 40dk \geq 4d$ and $\ell \coloneqq 2k(r+1)$.

 By Corollary \ref{cor:single-neighbor-layered-expander}, there is an $(\alpha,\gamma)$-single-neighbor $(\ell \times m)$-layered expander $G = (V,W,E)$ such that $\deg(w) = d+1$ for all $w \in W$.
 Let $V_0,\dots,V_\ell$ and $W_1,\dots,W_\ell$ denote the layers of $G$.
 Also fix some arbitrary element $x_\ell \in V_\ell$.
 We define
 \[\mathcal C \coloneqq \mathcal C_G \cup \{(\{x\},0) \mid x \in V_0\} \cup \{(\{x_\ell\},1)\}.\]
 Note that $\mathcal C$ is a set of XOR-constraints over $V$ of arity at most $d+1 = \ell_{\sf lo}$.

 To complete the proof, we show that $\mathcal C$ has the desired properties.
 First,
 \[|V| = (\ell + 1)m = (2k(r+1) + 1)40dk \leq 8kr \cdot 40dk = 320R_0 (\ell_{\sf hi} + 1)^{2}r \leq \delta \cdot \ell_{\sf hi}^{2} \cdot r\]
 for some suitable absolute constant $\delta$.
 Moreover, Falsifier wins the $\ell_{\sf lo}$-pebble game $\mathcal G_{\ell_{\sf lo}}(V,\mathcal C,\emptyset)$ by Lemma \ref{la:falsifier-win}.
 Finally, by Lemma \ref{la:verifier-win-few-rounds}, Verifier wins $\mathcal G_k^{r}(V,\mathcal C \setminus \{(\{x_\ell\},1)\},\beta_\ell)$ where $\beta_\ell\colon\{x_\ell\} \rightarrow \{0,1\}$ is the partial assignment defined via $\beta_\ell(x_\ell) = 1$.
 So Verifier wins the $r$-round $\ell_{\sf hi}$-pebble game $\mathcal G_{\ell_{\sf hi}}^{r}(V,\mathcal C,\emptyset)$ by Lemma \ref{la:strategy-assign-one-variable}.
\end{proof}

\section{Introduction}
\label{sec:introduction}
\input{introduction}

\section{Preliminaries}
\label{sec:preliminaries}
\input{preliminaries}

\section{Upper Bounds}
\label{sec:upper-bounds}
\input{upper-bounds}

\section{Long Sequences of Stable Colorings}
\label{sec:long-sequences}
\input{long-sequences}

\section{Lower Bounds on the Iteration Number of WL}
\label{sec:lower-bounds}
\input{lower-bounds}

\section{Trading Variable Number for Quantifier Depth}
\label{sec:tradeoffs}
\input{tradeoffs}

\section{Conclusion}
\label{sec:conclusion}
\input{conclusion}

\bibliographystyle{plainurl}
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redpajama_set_name: RedPajamaArXiv